Three fair dice are cast. In 10 independent casts, let X be...

Three fair dice are cast. In 10 independent casts, let $X$ be the number of times all three faces are alike and let $Y$ be the number of times only two faces are alike. Find the joint $\mathrm{pmf}$ of $X$ and $Y$ and compute $E(6 X Y)$.

Key Concepts

Combinatorial Analysis

Combinatorial analysis involves counting methods that determine the number of ways outcomes can occur. In problems dealing with dice or similar experiments, combinatorial techniques are essential for calculating the probabilities of complex events by accurately enumerating favorable and total outcomes.

Expectation and Linearity of Expectation

The concept of expected value provides a measure of the central tendency or average outcome for a random variable. The linearity of expectation, which holds regardless of the dependence between random variables, permits the computation of expectations of sums or products (when properly handled) by decomposing them into simpler components. This is critical when assessing quantities such as E(6XY) in joint distributions.

Joint Probability Mass Function (PMF)

The joint PMF describes the probability that two discrete random variables simultaneously take on specified values. It captures the behavior of the pair as a whole rather than just individually, which is particularly useful when the two random variables may be related by a common experiment or event outcomes.

Multinomial Distribution

When multiple mutually exclusive outcomes are observed over a fixed number of independent trials, the counts of these outcomes follow a multinomial distribution. This framework generalizes the binomial distribution to more than two possible outcomes per trial, allowing for a systemic study of events where each trial can result in one of several outcomes.

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